We consider unsupervised representation learning from i.i.d. data \(\mathcal {D}\!=\!\lbrace \mathbf {x}_n\rbrace ^{N}_{n=1}\)  consisting of observations \(\mathbf {x}_n \in \mathbb {R}^{D}\) . We also define a continuous latent space such that representations \(\mathbf {z}_n\!\in \!\mathbb {R}^{K}\) . Similar to aes [1]}, we consider two neural networks \(e_{{\psi }}:\mathbb {R}^{D}\)\(\rightarrow \)\(\mathbb {R}^{K}\)  and \(d_{{\phi }}\)\(:\mathbb {R}^{K}\)\(\rightarrow \)\(\mathbb {R}^{D}\) , widely known as the encoder and decoder. These have parameters \({\theta }=\lbrace {\psi },{\phi }\rbrace \) . We refer to the composition of encoder and decoder as \(f_{\theta }= d_{\psi }\circ e_{\phi }\) .
